Lemma 3.2 ([1]}, Lemma 2.2)  
Let \(\lbrace X_i\rbrace _{i\in [n]}\)  be the number of balls in bins \(i\in [n]\)  when \(\lfloor \theta n\rfloor , 0\le \theta \le 1\) , balls are independently and uniformly placed into \(n\)  bins. For \(k\in \lfloor \theta n\rfloor \)  and \(S\subseteq [n]\) , we have
\({\mathbb {P}}\left(\max _{i\in S}X_i<k\right)\le 2\exp \left(-\frac{\theta ^k|S|}{ek!}\right).\) 

